Figure 1.
Graphs of the discrete solutions with different values of $\tau$ . The smaller jumps of $u^\tau$ correspond to the larger parameter $\beta$
-
Previous Article
Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition
- NHM Home
- This Issue
-
Next Article
Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
September
2018, 13(3): 423-448.
doi: 10.3934/nhm.2018019
Perturbations of minimizing movements and curves of maximal slope
Via della Ricerca Scientifica 1, Rome, 00133, Italy |
We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.
Keywords:
Minimizing movements,
gradient flows,
perturbations,
homogenization theory,
multi-well potentials.
Mathematics Subject Classification:
47J35, 47J30, 35B27, 35K90, 49J05.
Citation:
Antonio Tribuzio. Perturbations of minimizing movements and curves of maximal slope. Networks & Heterogeneous Media,
2018, 13
(3)
: 423-448.
doi: 10.3934/nhm.2018019
![]()
References:
| [1] |
N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in
Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16
(Springer), 30 (2017), 139-151.
doi: 10.1007/978-981-10-6283-4_12. |
| [2] |
N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885.Google Scholar |
| [3] |
F. Almgren, J. E. Taylor and L. Wang,
Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 31 (1993), 387-438.
doi: 10.1137/0331020. |
| [4] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhauser Verlag, second edition, 2008. |
| [5] |
A. Braides,
Γ-convergence for Beginners, Oxford University Press, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
| [6] |
A. Braides,
Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014.
doi: 10.1007/978-3-319-01982-6. |
| [7] |
A. Braides, M. Colombo, M. Gobbino and M. Solci,
Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689.
doi: 10.1016/j.crma.2016.04.011. |
| [8] |
M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow,
Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp.
doi: 10.1142/S0218202512500170. |
| [9] |
L. C. Evans,
Partial Differential Equations, American Mathematical Society, second edition, 2010.
doi: 10.1090/gsm/019. |
| [10] |
F. Fleissner, Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016).Google Scholar |
| [11] |
F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi,
Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256.
doi: 10.2422/2036-2145.201711_008. |
| [12] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [13] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
show all references
References:
| [1] |
N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in
Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16
(Springer), 30 (2017), 139-151.
doi: 10.1007/978-981-10-6283-4_12. |
| [2] |
N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885.Google Scholar |
| [3] |
F. Almgren, J. E. Taylor and L. Wang,
Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 31 (1993), 387-438.
doi: 10.1137/0331020. |
| [4] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhauser Verlag, second edition, 2008. |
| [5] |
A. Braides,
Γ-convergence for Beginners, Oxford University Press, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
| [6] |
A. Braides,
Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014.
doi: 10.1007/978-3-319-01982-6. |
| [7] |
A. Braides, M. Colombo, M. Gobbino and M. Solci,
Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689.
doi: 10.1016/j.crma.2016.04.011. |
| [8] |
M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow,
Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp.
doi: 10.1142/S0218202512500170. |
| [9] |
L. C. Evans,
Partial Differential Equations, American Mathematical Society, second edition, 2010.
doi: 10.1090/gsm/019. |
| [10] |
F. Fleissner, Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016).Google Scholar |
| [11] |
F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi,
Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256.
doi: 10.2422/2036-2145.201711_008. |
| [12] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [13] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
Figure 2.
Pinned motion produced by perturbations diverging in the interval $(1, 2)$
Figure 3.
The graphs represent two discrete solutions for the same value of $\tau$ , corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear
Figure 4.
On the left the time chart of $u^\tau$ , on the right the plot of $\phi(u^\tau)$ , corresponding to perturbations with $\delta = 1$ . Note how the motion exits from the lower energy state when $t = 1$
Figure 7.
Graph for $\delta = 1$ . The motion always passes to the very next potential well
Figure 8.
Graphs of a discrete solution passing through two potential wells at every jump discontinuity
| [1] |
Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657 |
| [2] |
Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153 |
| [3] |
Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879 |
| [4] |
Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501 |
| [5] |
Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43 |
| [6] |
Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345 |
| [7] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
| [8] |
Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028 |
| [9] |
Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks & Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233 |
| [10] |
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025 |
| [11] |
Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295 |
| [12] |
Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157 |
| [13] |
Maroje Marohnić, Igor Velčić. Homogenization of bending theory for plates; the case of oscillations in the direction of thickness. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2151-2168. doi: 10.3934/cpaa.2015.14.2151 |
| [14] |
Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 |
| [15] |
Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 |
| [16] |
Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117 |
| [17] |
Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793 |
| [18] |
Dorin Ieşan. Strain gradient theory of porous solids with initial stresses and initial heat flux. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2169-2187. doi: 10.3934/dcdsb.2014.19.2169 |
| [19] |
Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073 |
| [20] |
Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573 |
2018 Impact Factor: 0.871
Tools
Metrics
Other articles
by authors
[Back to Top]